PREFACE TO THE SECOND EDITION FOREWORD |

CHAPTER ONE. MECHANISMS OF CRACK FORMATION AND GROWTH |

| 1. Fracture mechanics and strength of solid bodies |

| 2. Stress and displacement fields in the vicinity of a crack tip |

| 3. The Griffith-Orowan-Irwin concept. Steady and unsteady crack development |

| 4. The integral variational principle in the crack theory |

| 5. Modified Griffith criterion |

| 6. Variation of the total energy of the system with small increments of the crack length |

| 7. Model of a crack with a thin plastic zone (delta_{k}-model) |

| 8. Invariant *J*-integral |

| 9. Invariant Г-integrals |

| 10. Conditions for local fracture of piezoelectrics (ПК-criteria) |

| 11. Some alternative criteria of crack development |

CHAPTER TWO. STRESS INTENSITY FACTOR |

| 12. The stress intensity factor as a fundamental characteristic of a crack containing material |

| 13. Finite element approach in fracture mechanics |

| 14. Method of the boundary integral equations in fracture mechanics |

| 15. Examples of computation of stress intensity factors by both finite element method and method of boundary integral equation |

| 16. Method of sections for an approximate calculation of the stress intensity factor |

| 17. Some properties of materials characterizing resistance to crack propagation |

CHAPTER THREE. CRACKS IN ELASTIC PLASTIC BODIES |

| 18. Solutions of selected plane and three-dimensional
problems |

| 19. Torsion and tension of a cylinder containing an external
annular crack |

| 20. Structural means for crack arrests (stiffening ribs) |

| 21. Structural methods for crack arrest (repair patches,
load relief holes) |

| 22. Crack systems |

| 23. Composite elastic bodies |

| 24. Methods of crack trajectory analysis |

CHAPTER FOUR. CRACKS IN ELASTIC-PLASTIC BODIES |

| 25. Experimental study of the plastic zone at the slit-crack tip |

| 26. The plastic zone analysis |

| 27. Critical state of crack-containing plane and space |

| 28. Eastic-plastic problem for a plane containing rectilinear slits |

| 29. Prior-to-critical crack growth |

| 30. Low-cyclic fatigue life |

| 31. Experimental results of single and cyclic loading |

| 32. Life analysis of structural elements based on number of cycles |

| 33. Toughness limit |

| 34. Stress analysis based on the toughness limit |

| 35. Examples of strength analysis with allowance for a crack |

CHAPTER FIVE. CRACKS IN LINEAR VISCOELASTIC SOLIDS |

| 36. Viscoelastic analogue to Griffith--Sack problems |

| 37. Viscoelastic-plastic analogues to Griffith and Sack problems |

| 38. Material sensitivity to rate and frequency of load application |

| 39. Delayed fracture of polymer materials |

| 40. Study of crack growth in polymers under tension |

APPENDIX REFERENCES |

SUPPLEMENTAL REFERENCES INDEX |

Preface to the second edition

From ancient times, man has been concerned with determining both the strength
of certain materials and the cause of their fracture. However, over the
centuries, knowledge of the strength of materials and the nature of fracture
was accumulated only sporadically, having been passed down from generation to
generation in the framework of an art or skill rather than as a science.

Even today, the true nature of fracture is not completely understood. The
increasing number of catastrophic failures of ships, aircraft, and rockets as
caused by a sudden fracture or crack propagation has demonstrated the
inadequacy of currently-used methods of analysis. Indeed, these dramatic
failures have served to place the study of fracture in the forefront of
advanced engineering research.

Today, scientists are studying the phenomenon of fracture from the perspective
of solid body mechanics. The aim of these studies is to describe the major
features of fracture in various groups of materials by means of
rigorously-formulated models. The use of the fundamental principles and methods
of solid body mechanics when studying the fracture process has led to the name
of this new science, which is called *fracture mechanics*.

In a broad sense, fracture mechanics is that portion of the science of strength
of materials which is concerned with analyzing the carrying capacity of a body
either with or without existing cracks. Of necessity, fracture mechanics also
considers the process of crack formation and growth.

Galileo must be given credit as a founder of fracture mechanics. He was the
first to find that the breaking load of a bar in tension is proportional to its
cross-sectional area, and is independent of its length. The work of Galileo and
others -- such as Hooke, Coulomb, Saint Venant, and Mohr -- is generally considered
the first step in the study of fracture mechanics. Their findings, characterized by both an
extensive investigation of deformation and the development of various fracture
criteria (having phenomenological implications), are referred to as strength
theories. The essence of these theories is that a fracture occurs at the moment
when, at some point in a body, such quantities as stress and strain (or their
combination) reach a critical level. These approaches ignore the process of
crack propagation in the body, and are justified only in cases where the
development of cracks (and the attendant loss in carrying capacity) takes place
in a relatively small zone of the critical region.

If a strength analysis were performed today, an appropriate strength theory
would be used that takes into account criteria such as a maximum principal
stress, shear stress, or octahedral stress, depending on the type of material
and the operating conditions of the structure. Such an analysis would still be
insufficient, and will continue to be so even if an allowance is made for
future improvements in strength theories. For example, problems concerning the
equilibrium of crack-containing elastic bodies are of particular importance.
However, general solutions to these problems based on strength theory are
associated with enormous mathematical difficulties. Moreover, these solutions
contain much more information than is needed. What is really needed is to
determine whether a body has sufficient capacity at a given load. That is, the
major object of interest is not in a general solution to a complex problem of
equilibrium of a cracked body, but rather the existence or nonexistence of
a particular solution at a given load. In other words, from a mathematical point
of view, fracture occurs with limit states that ensure the nonexistence of
strength-theory solutions. These limit states are integral characteristics of
the fracture process, and are in agreement with the general global concept of
fracture of solid bodies.

When the phenomenological strength-theory approach is used to develop models
for crack development in solids, it is most common to initially assume some
disturbance in the form of a pre-existing (original) crack. This assumption is
consistent with the observed presence of material imperfections, such as cracks
that occur during the fabrication process of a particular structure. Hence,
when deriving various strength criteria based on the process of fracturing,
relations are obtained that are in formal agreement with regular strength
criteria, differing only in that the derived relations involve constants that
depend on the coordinates, lengths, and geometry of the pre-existing cracks.

Current research in the field of fracture mechanics is aimed at more than the
carrying-capacity concept. In fact, the investigation of the fracture process
now represents an independent field of interest. Monitoring the process of
fracture, and learning the laws that govern it are of great practical
importance. For example, in operating structures it is essential to suppress
the process of fracture, whereas when cutting metals, it is desirable to
facilitate rupture.

This book presents the fundamentals of the mechanics of crack development in
solids, and some special problems of fracture mechanics involving higher levels
of mathematics.

Part One is devoted to the fundamental concepts and methods concerning elastic
and elastic-plastic fracture. Examples are given that demonstrate the results
of using various fracture criteria to determine both the critical and allowable
crack lengths at static and cyclic loadings. For the purposes of independent study or for
delivering a short course, Sections 1 to 3, 12, 16, 17, 25, 30, 33, and 34 are
recommended.

Part Two is a systematic presentation of several problems of fracture mechanics
such as the effects of hydrogen-containing materials, stress corrosion, thermal
and dynamic loadings, and electromagnetic interaction in piezoelectric
materials. Solutions of such problems involve complex mathematics, in which
readers should be adequately grounded.

Fracture mechanics problems are so extensive that some topics are not covered;
these include the fracture of plates and shells, and methods of experimental
determination of fracture toughness. It is felt that these topics are
sufficiently represented in related publications.

This second edition of the book, which contains both revisions and additional
material, can now be used as a text for graduate students in universities.

The authors express their deep gratitude to Dr. N.Robes for his thorough review
and comments, which contributed to the further improvement of this edition.

The term *fracture mechanics* is somewhat unsettling to many people. This
is because, until recently, the major emphasis in mechanics was on the strength
and resistance of materials. To speak of fracture is as uncomfortable for some
as it is to speak of a fatal illness. However, just as in preventing a fatal
disease, one must know its nature, symptoms, and behavior; to ensure the
strength of a structure, one must be aware of the causes and nature of its
potential failure.

The problem of fracture is vital in the science of strength of materials.
However, not only has fracture mechanics, as an independent branch of the
mechanics of deformable solids, originated quite recently, but its boundaries
are not yet clearly defined. Therefore, it is of paramount interest to combine
the efforts of representatives from many different branches of science and
engineering for a complete study of the fracture concept. It is also important
that differences in terminology (that are usual for different sciences), and
the widespread conviction that the solution to everything lies in a particular
portion of the general problem, do not lead to a situation in which disputes
about the concepts are replaced by arguments about the words.

At present, routine fracture mechanics is the study of conditions under which
a crack or a system of cracks undergoes propagation. However, cracks are of
different natures, and are considered on different scale levels. The case on
one extreme is the fracture of a crystal grain, which initiates with
a submicroscopic crack when two atomic layers move apart by such a distance that
the forces of interaction between the atoms may be neglected. An example of the
other extreme is a crack occurring in a welded turbine rotor in a nuclear
reactor, when the crack's length and width may amount to centimeters; this is
referred to as a macroscopic fracture.

In the first case, the condition for crack propagation is defined by the
configuration of atoms at the crack tip. Considered here is a discrete crystal lattice
formed by atoms rather than a continuous medium; therefore, the very concept of
the "crack tip" becomes uncertain. The study of this kind of submicroscopic
crack and its behavior in interaction with other lattice defects is,
essentially, in the province of solid-state physics rather than mechanics;
however, the methods of classical theory of elasticity are fully applicable to
problems of this nature. The line between modern physics and mechanics is not
well defined; nevertheless, it must be drawn to avoid possible terminological
confusion.

A macroscopic fracture has dimensions exceeding by several orders the size of
the largest structural constituent of the material (the constituent must
contain a sufficient number of crystal grains for its properties not to differ
from those of any other element of similar size which may be isolated from the
material). It is precisely this condition that makes it possible to solve such
a crack problem within the framework of mechanics of a solid body. The
formulated condition refers to an ideal situation in order to make the theory
applicable; in real conditions one may depart from this stringent requirement,
but this in no way makes the theory groundless. Assuming the material to be
continuous, homogeneous, and elastic, and using the techniques of the classical
theory of elasticity, we inevitably arrive at the paradoxical conclusion that
the stresses grow infinitely near the crack tip. This paradox is a sort of
penalty paid for the simplicity associated with using the linear theory of
elasticity in a region where its application is know to be invalid.

So-called linear fracture mechanics assume that a physically impossible
singularity is a reality. Such an approach is not new and not so unusual for
continuum mechanics; recall, for example, the vortex filaments with zero cross
section and finite circulation. It appears that the work of crack propagation,
which is done either as a result of increase of external forces or reduction of
the elastic energy of the body with the crack size increase, is expressed
directly through the coefficient of the singular term in the formula for
stress. This coefficient is referred to as the stress intensity factor, and is
of fundamental importance for the entire theory. The work of crack propagation
may be associated with overcoming the forces of surface tension (Griffith's
concept), or the plastic deformation in the small region of the immediate
neighborhood of the crack tip, or other physical causes. The factor to be
emphasized is that the size of the region, where the laws of the linear theory
of elasticity are in some way violated, must be very small. The ability of the
crack to further propagate is then determined by the sole characteristic: the
work per unit length of the propagation path, or the critical stress intensity
factor.

If the size of the zone, where the relations of the linear theory of elasticity
are violated, is large, one should consider the laws of nonlinear fracture
mechanics. It appeared at the beginning that formal indifference of linear
fracture mechanics to both the object and the scale, mathematical equivalence
of problems associated with entirely different physical phenomena, would make
it possible to establish nonlinear mechanics in a similar uniform manner. It
was later found to be quite different.

The principal problem, on which the efforts of scientists have been focused in
recent years, concerns the conditions of either equilibrium or the propagation
of a large crack in a sufficiently plastic material. Scientists have been involved
in the theory and practical applications of fracture mechanics for evaluating
the strength of large-scale structural elements. They have shown that the
plastic zone ahead of a crack is sufficiently extensive so that the macroscopic
theory of plasticity, which assumes that the medium is continuous and
homogeneous, holds good. For the plane state of stress, the
Leonov--Panasyuk--Dugdale model, which substitutes the plastic zone by
a no-thickness segment extending the crack, appears to be satisfactory. In
particular, this book presents an analysis of the corresponding elastic-plastic
problem that is solved numerically by using the finite element method (FEM).
The presented FEM solution confirms the validity of the model used: as the
number of elements in the model is increased, the plastic zone contracts; in
the limit state, the plastic zone is expected to degenerate into a segment
when, with the infinite refinement of the mesh, the solution approaches the
exact result. Many authors, when considering submicroscopic cracks at the
atomic scale level, assume the hypothesis that nonlinear behavior in
interaction between atoms is significant only within a single interactomic
layer (similar to the computation of the so-called Peierls dislocation). Again,
as in the linear theory, the analogy is merely formal; in this case, it is of
an artificial nature, and the judgments about the relative validity of the
model in various cases are based on entirely different considerations; the
persuasiveness of argumentation in favor of this model appears to vary in
a wide range.

Unfortunately, a plane state of stress can never been realized in practice; at
a distance from the crack tip of approximately the thickness of a sheet, the
state of stress is essentially three-dimensional and far too complex for
analysis. In the case of plane strain, the shape of the plastic zone appears to
be different: it spreads transversely rather than longitudinally, and the model
of the plastic segment assumed for plane stress in no way reflects the reality.

The situation faced by a design engineer is complicated. The size of the
plastic zone in structural elements made of modern alloys is of the same order
as the thickness of the element; consequently, the state of stress throughout
the plastic region is essentially three-dimensional. Also, the most common
structural materials, carbon and alloy steels, are quite ductile. Crack
propagation begins when the plastic deformation encountered in the vicinity of
its tip becomes extensive, amounting to the order of tens of percent. The tip
of an originally sharp, say, fatigue, crack becomes blunt. Its flanks, which
were initially closed, separate transversely in a parallel fashion by a finite
distance, and further fracture takes place only when the opening reaches
a certain critical value. Thus, the theory of crack propagation in ductile
materials includes at least two elements: 1) the solution of an elastic-plastic
problem by taking into consideration both the finiteness of the plastic strain
and the boundary conditions over the deformed boundary, and 2) determination of
the condition of macrocrack formation in material which has undergone
a significant deformation accompanied by accumulation of microdefects.

The book by V.Z.Parton and E.M.Morozov is the first Russian monograph on the
above-discussed subject. It is based mainly on the results obtained by the
authors during their original research, and concerns the problems of nonlinear
fracture mechanics. It presents some elastic-plastic problems for crack-containing bodies.
The greater part of the book is devoted to linear fracture mechanics and also
to some new developments in this field which lead to governing equations that
may be nonlinear.

In spite of certain limits imposed by linear fracture mechanics, a wide variety
of problems may be reliably solved using its methods. Development of this
theory is focused on accumulating data from already solved elasticity problems
concerning cracks of various shape in various bodies. The amount of such
information continually grows both abroad and in the USSR. Many results
obtained by foreign authors became available by means of numerous translations
of books and published articles. In particular, a translation of the
seven-volume advanced treatise "Fracture," edited by H.Liebowitz (USA) [247],
is now being published.

The present book may be considered as a significant contribution to the
database of fracture mechanics. Some features of the book deserve special
mention. First of all, it is the new variational principle that makes it
possible to approximately solve numerous problems; in particular, to find the
trajectory of crack propagation in a nonuniform stress field. Also,
a straightforward approach for an approximate determination of the stress
intensity factor is included; it enables one to obtain a reasonable evaluation
for those cases where an exact solution of the elasticity problem is
impossible, and the numerical computation is extremely laborious. In addition,
a series of newly solved dynamic problems for bodies subjected to cyclic
(periodic) loading is provided.

Linear fracture mechanics has been developed by applying its concepts to the
problems of crack-growth kinetics as a function of either time or, in the case
of fatigue fracture, the number of cycles. It should be noted that the
kinetics, both linear and nonlinear, are presumed to be essentially local: all
fracture processes regardless of their nature are assumed to occur in an end
region of very small size; outside this region the material is in an elastic
state. Then, the stress intensity factor becomes the only representative of the
state of stress in any kinetic equations. The chapters of books concerned with
fatigue fracture are structured following this approach.

In conclusion we shall note that significant advancement in the field of the
mechanics of crack propagation has led to such a wide-spread perception that it
reflects the entire fracture mechanics. However, the subject matter of fracture
mechanics should be understood in a much wider context. For instance, in metals
loaded at high temperature, the fracture is of a scattered nature: micropores
are accumulated at the grain boundaries over the entire volume of the body,
followed by their merging and, finally, combining to form a macrocrack. The
macrocrack is merely the final, visible result of the damage accumulation
process that cannot be recognized by the naked eye, but can well be seen using
appropriate optical devices. A similar character of fracture is apparently
observed in some polymers, but in this case a more precise technique is needed
to detect the microdamage.

The importance of statistics in evaluating the strength of structures is widely
known. The statistical theory of fracture must also be considered as an
inherent part of fracture mechanics. It should be mentioned, however, that the
sophisticated and theoretical probabilistic analysis is often used with rather
primitive mechanical modelling. This, no doubt, may be explained by the
complexity of the studied subject.

The book elaborates on an extremely important problem. It contains a number of
interesting remarks and considerations, sometimes of a tentative nature, that
fuel thinking and stimulate further work.

Yu.N.Rabotnov, Academician, USSR Academy of Sciences

**Vladimir Zalmanovich Parton** (1936-2001)

Prof. V.Z.Parton was Head of Department of Higher Mathematics at the Institute of Chemical Engineering
of Moscow. He was a well-known specialist
in elasticity theory and fracture mechanics. He was a co-author of the books
"Mathematical methods of the theory of elasticity" (1981), "Dynamics of brittle
fracture" (1988), as well as the author of "Fracture Mechanics:
from theory to practice" (1990; second edition, URSS, 2007).

**Evgenii Mikhailovich Morozov**

Prof. E M.Morozov works at the Department of Strength Physics of the Moscow Institute of Physics and Engineering. He
is a recognized specialist in fracture mechanics and strength of solid
bodies. Also, he was one of the pioneers of fracture mechanics
in Russia. Prof. Morozov received the award of the Council
of Ministers of the Soviet Union and the title of Honoured Worker
of Science of the Russian Federation. E. M. Morozov's name has been
included in various Russian and English "Who's who" handbooks.
He is the author of many books and manuals on fracture mechanics.